Priors

\(\alpha\)

tibble(x = seq(0, 1.3, length = 10^(5)),
       y = with(prior_params, gamma_density(x,
                                     mean = alpha_mean,
                                     sd = alpha_sd,
                                     bounds = alpha_bounds))) %>%
   ggplot(aes(x=x, y = y)) +
  geom_line(alpha = .8) + 
  geom_ribbon(aes(x=x,ymin=0,ymax=y),
              fill="black",
              alpha=.6) +
  theme_c(legend.text=element_text(size = 10)) +
  viridis::scale_fill_viridis(discrete=TRUE,
                             option = "rocket", begin=.3,end=.8) +
  labs(x = "Value",
       y = "Probability Density", 
       title = latex2exp::TeX("Definition of Prior for $\\alpha$"),
       fill ='',
       subtitle = paste0("Mean: ", prior_params$alpha_mean,
                         ", SD: ", prior_params$alpha_sd))

\(\beta\)

tibble(x = seq(0, 1, length = 10^(5)),
       y = with(prior_params, beta_density(x,
                                     mean = beta_mean,
                                     sd = beta_sd,
                                     bounds = beta_bounds))) %>%
   ggplot(aes(x=x, y = y)) +
  geom_line(alpha = .8) + 
  geom_ribbon(aes(x=x,ymin=0,ymax=y),
              fill="black",
              alpha=.6) +
  theme_c(legend.text=element_text(size = 10)) +
  viridis::scale_fill_viridis(discrete=TRUE,
                             option = "rocket", begin=.3,end=.8) +
  labs(x = "Value",
       y = "Probability Density", 
       title = latex2exp::TeX("Definition of Prior for $\\beta$"),
       fill ='',
       subtitle = paste0("Mean: ", prior_params$beta_mean,
                         ", SD: ", prior_params$beta_sd))

\(P(S_1|\text{untested})\)

tibble(x = seq(0, 1, length = 10^(5)),
       y = with(prior_params, beta_density(x,
                                     mean = s_untested_mean,
                                     sd = s_untested_sd,
                                     bounds = s_untested_bounds))) %>%
   ggplot(aes(x=x, y = y)) +
  geom_line(alpha = .8) + 
  geom_ribbon(aes(x=x,ymin=0,ymax=y),
              fill="black",
              alpha=.6) +
  theme_c(legend.text=element_text(size = 10)) +
  viridis::scale_fill_viridis(discrete=TRUE,
                             option = "rocket", begin=.3,end=.8) +
  labs(x = "Value",
       y = "Probability Density", 
       title = latex2exp::TeX("Definition of Prior for $P(S_1|untested)$"),
       fill ='',
       subtitle = paste0("Mean: ", prior_params$s_untested_mean,
                         ", SD: ", prior_params$s_untested_sd))

\(P(S_0| \text{test}_+, \text{untested})\)

tibble(x = seq(0, 1, length = 10^(5)),
       y = with(prior_params, beta_density(x,
                                     mean = p_s0_pos_mean,
                                     sd = p_s0_pos_sd,
                                     bounds = p_s0_pos_bounds))) %>%
   ggplot(aes(x=x, y = y)) +
  geom_line(alpha = .8) + 
  geom_ribbon(aes(x=x,ymin=0,ymax=y),
              fill="black",
              alpha=.6) +
  theme_c(legend.text=element_text(size = 10)) +
  viridis::scale_fill_viridis(discrete=TRUE,
                             option = "rocket", begin=.3,end=.8) +
  labs(x = "Value",
       y = "Probability Density", 
       title = latex2exp::TeX("Definition of Prior for $P(S_0|test_+,untested)$"),
       fill ='',
       subtitle = paste0("Mean: ", prior_params$p_s0_pos_mean,
                         ", SD: ", prior_params$p_s0_pos_sd))